I had an annoying problem where my Lync client was displaying a blank screen on the stage whenever someone else shared their desktop (or any other application) with me. Incidentally, in reverse when I shared my desktop with the other person, they could see my desktop just fine.
After many hours of troubleshooting trying various things from upgrading video drivers to uninstalling programs such as Cisco Webex Productivity Tools, I found this article: http://blogs.technet.com/b/lync_tips_and_tricks/archive/2013/01/11/lync-2013-desktop-sharing-shows-blank-screen.aspx. The author explains the symptoms accurately and the suggested resolution fixed my problem. For convenience, I have provided the fix below.
Open registry editor and remove the following two keys:
HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Internet Explorer\ActiveX Compatibility\{00000000-0000-0000-0000-000000000000} (if using LYNC 2013 32 bits on Windows 8 64 bits)
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Internet Explorer\ActiveX Compatibility\{00000000-0000-0000-0000-000000000000} (if running LYNC 2013 64 bits or running LYNC 2013 32 bits on Windows 8 32 bits)
After deleting the keys, my problems with desktop sharing were resolved.
An alternative resolution method was mentioned in the comments of the article as well that is worth sharing. Try changing your default web browser to something other than Internet Explorer (IE) such as Google Chrome or Mozilla Firefox. Test desktop sharing again. I am told that this resolves the issue as well, although I have not personally tested it.